Calculus 8th Edition

Published by Cengage
ISBN 10: 1285740629
ISBN 13: 978-1-28574-062-1

Chapter 16 - Vector Calculus - 16.9 The Divergence Theorem - 16.9 Exercises - Page 1186: 18


$\dfrac{3 \pi}{2}$

Work Step by Step

Divergence Theorem: $\iiint_Ediv \overrightarrow{F}dV=\iint_S \overrightarrow{F}\cdot d\overrightarrow{S} $ where, $div F=\dfrac{\partial P}{\partial x}+\dfrac{\partial Q}{\partial y}+\dfrac{\partial R}{\partial z}=\dfrac{\partial (z \tan^{-1} y^2)}{\partial x}+\dfrac{\partial (z^3(\ln (x^2+1))}{\partial y}+\dfrac{\partial (z)}{\partial z}=1$ Divergence Theorem: $\iiint_Ediv \overrightarrow{F}dV=\iint_S \overrightarrow{F}\cdot d\overrightarrow{S} $ $=(2 \pi) \times \int_{0}^{1} [zr] dr$ $=(2 \pi) \times \int_0^{1} (r-r^3) dr$ or, $=\dfrac{\pi}{2}$ The flux through the disk is given by: $-\iint_{D} z dA=-\pi(1)^2=-\pi$ Now, the flux through paraboloid is equal to the (total Flux -Flux Through Disk) , that is, $=\dfrac{\pi}{2}-(-\pi)=\dfrac{3 \pi}{2}$
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